The Majorization Theorem and Signless Dirichlet Spectral Radius of Connected Graphs

Abstract

Let \(D(G)\) denote the signless Dirichlet spectral radius of the graph \(G\) with at least a pendant vertex, and \(\pi_1\) and \(\pi_2\) be two nonincreasing unicyclic graphic degree sequences with the same frequency of number \(1\). In this paper, the signless Dirichlet spectral radius of connected graphs with a given degree sequence is studied. The results are used to prove a majorization theorem of unicyclic graphs. We prove that if \(\pi_1 \unrhd \pi_2\), then \(D(G_1) \leq D(G_2)\) with equality if and only if \(\pi_1 = \pi_2\), where \(G_1\) and \(G_2\) are the graphs with the largest signless Dirichlet spectral radius among all unicyclic graphs with degree sequences \(\pi_1\) and \(\pi_2\), respectively. Moreover, the graphs with the largest signless Dirichlet spectral radius among all unicyclic graphs with \(k\) pendant vertices are characterized.